关于共形域论中一个泛函-差分算子的spectral theory
On the spectral theory of one functional-difference operator from conformal field theory
摘要：
本文考虑一个泛函-差分算子$H=U+U^{-1}+V$，其中$U$和$V$是满足$UV=q^{2}VU$且具有$q=e^{πiτ}$和$τ>0$的自伴Weyl算子.算符$H$在共形域论和量子群的表示论中都有应用。使用模量子双对数-一个$q$ -变形的欧拉双对数-我们定义的散射解决方案和Jost解决方案，推导出一个明确的公式，在希尔伯特空间$L^{2}(\mathbb{R})$的自伴算子$H$的预解式，并证明了本征函数展开定理。后者是一个$q$ -变形著名的Kontorovich-Lebedev变换理论中的特殊功能。我们还提出了一个公式的散射理论的运营商$H$。
英文摘要：
In the paper we consider a functional-difference operator $H=U+U^{-1}+V$, where $U$ and $V$ are self-adjoint Weyl operators satisfying $UV=q^{2}VU$ with $q=e^{πiτ}$ and $τ>0$. The operator $H$ has applications in the conformal field theory and in the representation theory of quantum groups. Using modular quantum dilogarithm - a $q$-deformation of the Euler's dilogarithm - we define the scattering solution and the Jost solutions, derive an explicit formula for the resolvent of the self-adjoint operator $H$ in the Hilbert space $L^{2}(\mathbb{R})$, and prove the eigenfunction expansion theorem. The latter is a $q$-deformation of the well-known Kontorovich-Lebedev transform in the theory of special functions. We also present a formulation of the scattering theory for the operator $H$.
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